This course will cover a few combinatorial concepts and tools that can be applied in
(algebraic) topology and group theory: Forman’s Morse theory for cell complexes (also known
as discrete Morse theory); shellability of posets, simplicial complexes, and regular CW
complexes; Garside monoids and Garside groups. In all these topics, the underlying idea is to
translate topological or algebraic problems in a purely combinatorial language.
See the web page of the 2024/2025 PhD programme for the schedule.
Topics
- Partially ordered sets (posets)
- Cell complexes and simplicial complexes
- Discrete Morse theory
- Shellability of regular CW complexes
- Lexicographic shellability of posets
- Combinatorial Garside structures, Garside monoids, and Garside groups
- Applications to the symmetric group and the braid group
Bibliography
- Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
- Batzies, Discrete Morse Theory for Cellular Resolutions
- Björner, Posets, Regular CW Complexes and Bruhat Order
- Björner-Brenti, Combinatorics of Coxeter groups
- Brown, Topology and Groupoids
- Charney-Meier-Whittlesey, Bestvina's normal form complex and the homology of Garside groups
- Dehornoy, Groups with a complemented presentation
- Dehornoy-Digne-Godelle-Krammer-Michel, Foundations of Garside Theory
- Dehornoy-Paris, Gaussian groups and Garside groups, two generalisations of Artin groups
- Forman, Morse Theory for Cell Complexes
- Fritsch-Piccinini, Cellular structures in topology
- Hatcher, Algebraic Topology
- Kozlov, Combinatorial Algebraic Topology
- McCammond, An Introduction to Garside Structures
- Stanley, Enumerative Combinatorics
- Wachs, Poset Topology: Tools and Applications
- Ziegler, Lectures on Polytopes
Notes